Cosine in Terms of Haversine

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Theorem

$\cos \theta = 1 - 2 \hav \theta$

where $\cos$ denotes cosine and $\hav$ denotes haversine.


Proof

\(\ds \hav \theta\) \(=\) \(\ds \dfrac 1 2 \paren {1 - \cos \theta}\) Definition of Haversine
\(\ds \leadsto \ \ \) \(\ds 2 \hav \theta\) \(=\) \(\ds 1 - \cos \theta\)
\(\ds \leadsto \ \ \) \(\ds \cos \theta\) \(=\) \(\ds 1 - 2 \hav \theta\)

$\blacksquare$


Sources